The intrinsic spatial resolution of modern whole-body PET scanners is about 4 mm. However, the practically achievable resolution of cardiac or abdominal PET imaging may be worse than 10 mm due to inevitable cardiac and respiratory motion. Motion artifacts from head motion are also one of the major hurdles in brain PET. Our work will allow dynamic as well as static imaging in the free- breathing patient while at the same time achieving the intrinsic resolution of PET. This ground- breaking (x2) improvement in spatial resolution will be accomplished while preserving, or sometimes increasing, the sensitivity of PET, which is quite unusual in medical imaging. The importance of the significantly improved accuracy, spatial resolution (i.e., from ~10 mm to ~4 mm) and sensitivity in PET enabled by this TR&D cannot be over emphasized because it will have significant impacts on many clinical applications, including: (1) revealing tumor heterogeneity (e.g., necrotic core), (2) imaging small lung tumors, (3) detecting non-transmural myocardial defects, and (4) staging and monitoring response to therapy in Alzheimer's disease using high-resolution PET imaging of tau. To achieve this goal, we propose novel MR-based PET motion correction and accurate/motion- dependent PET attenuation correction methods using PET/MR. Specifically, for conventional non- time-of-flight PET/MR, we propose to use a novel free-breathing ZTE/multi-echo sequence to obtain continuous attenuation coefficient maps of lungs, bones, fat and soft tissues. For the time-of-flight PET/MR, we propose a maximum a posteriori estimation of activity and attenuation correction factors (MAPAACF) method for robust attenuation coefficient estimation from TOF-PET data. We also propose novel and accurate MR-based motion estimation and tracking methods for imaging different organs with either rigid or non-rigid motion. Finally, we propose a novel low-rank tensor- based MR acceleration method that captures data correlation in multiple dimensions to significantly reduce MR imaging time.